1. Warm Up Determine if each statement is true or false. 1. The measure of an obtuse angle is less than 90°. 2. All perfect-square numbers are positive. 3. Every prime number is odd. 4. Any three points are coplanar. F T F T

2. 11. 5 PM28. $400 12. 4229. fraction – mult of 1/11 decimal – repeating mult. of 9 13. Hexagon31. 34, 55, 89, sum of previous 2 terms 14. n(n + 1)32. middle # is mean of the other 2 #s 15. N – 1 37. C 16. 2638. J 17. Y = - 1 39. D 18. X = -1 43. angles and segments are equal 19. m  1 = m  2 = 90°44. no 20. 256, 65,53646. yes 21. ½, 1/16, 1/3248. 4x 22. -15, 1850. 3x + 6 24. T52. (-1, 1); (0, 3); (4, 2) 25. F, n = 253. (3, -2); (4, 0); (8, -1) 26. F, 3-4-5 triangle 27. T

3. By phrasing a conjecture as an if-then statement, you can quickly identify its hypothesis and conclusion. Example 1 "A number is divisible by 3 if it is divisible by 6." Identify the hypothesis and conclusion of the statement. Hypothesis: A number is divisible by 6. Conclusion: A number is divisible by 3.

4. “If p, then q” can also be written as “if p, q,” “q, if p,” “p implies q,” and “p only if q.” Writing Math Many sentences without the words if and then can be written as conditionals. To do so, identify the sentence’s hypothesis and conclusion by figuring out which part of the statement depends on the other. Example 2A Write a conditional statement from the sentence “Two angles that are complementary are acute.” If two angles are complementary, then they are acute. Identify the hypothesis and the conclusion. Two angles that are complementary are acute.

5. Write a conditional statement from the following. Example 2B: Writing a Conditional Statement If an animal is a blue jay, then it is a bird. The inner oval represents the hypothesis, and the outer oval represents the conclusion.

6. A conditional statement has a truth value of either true (T) or false (F). It is false only when the hypothesis is true and the conclusion is false. To show that a conditional statement is false, you need to find only one counterexample where the hypothesis is true and the conclusion is false. Determine if the conditional is true. If false, give a counterexample. Example 3A: An even number greater than 2 will never be prime, so the hypothesis is false. 5 + 4 is not equal to 8, so the conclusion is false. However, the conditional is true because the hypothesis is false. If an even number greater than 2 is prime, then 5 + 4 = 8.

7. Example 3B: Determine if the conditional “If a number is odd, then it is divisible by 3” is true. If false, give a counterexample. An example of an odd number is 7. It is not divisible by 3. In this case, the hypothesis is true, but the conclusion is false. Since you can find a counterexample, the conditional is false. If the hypothesis is false, the conditional statement is true, regardless of the truth value of the conclusion. Remember!

8. The negation of statement p is “not p,” written as ~p. The negation of a true statement is false, and the negation of a false statement is true. DefinitionSymbols A conditional is a statement that can be written in the form "If p, then q." p  qp  q Related Conditionals

9. DefinitionSymbols The converse is the statement formed by exchanging the hypothesis and conclusion. q  pq  p Related Conditionals DefinitionSymbols The inverse is the statement formed by negating the hypothesis and conclusion. ~p  ~q Related Conditionals DefinitionSymbols The contrapositive is the statement formed by both exchanging and negating the hypothesis and conclusion. ~q  ~p Related Conditionals

10. Write the converse, inverse, and contrapostive of the conditional statement “If an animal is a cat, then it has four paws.” Find the truth value of each. Example 4 If an animal is a cat, then it has four paws. Inverse: If an animal is not a cat, then it does not have 4 paws. Converse: If an animal has 4 paws, then it is a cat. Contrapositive: If an animal does not have 4 paws, then it is not a cat; True. There are other animals that have 4 paws that are not cats, so the converse is false. There are animals that are not cats that have 4 paws, so the inverse is false. Cats have 4 paws, so the contrapositive is true.

11. Related conditional statements that have the same truth value are called logically equivalent statements. A conditional and its contrapositive are logically equivalent, and so are the converse and inverse.